The sum of two numbers is $107$, and their difference is $35$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 107}$ ${x-y = 35}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 142 $ $ x = \dfrac{142}{2} $ ${x = 71}$ Now that you know ${x = 71}$ , plug it back into $ {x+y = 107}$ to find $y$ ${(71)}{ + y = 107}$ ${y = 36}$ You can also plug ${x = 71}$ into $ {x-y = 35}$ and get the same answer for $y$ ${(71)}{ - y = 35}$ ${y = 36}$ Therefore, the larger number is $71$, and the smaller number is $36$.